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We prove the folklore endpoint multilinear -plane conjecture originating in a paper of Bennett, Carbery and Tao where the almost sharp multilinear Kakeya estimate was proved. Along the way we prove a more general result, namely the endpoint multilinear -variety theorem. Finally, we generalize our results to the endpoint perturbed Brascamp–Lieb inequalities using techniques in earlier sections.
We show -boundedness, for , of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought of as a discrete counterpart of Littlewood–Paley theory. It is a very robust approach which allows us to proceed as in the continuous case.
We study the Kato problem for divergence form operators whose ellipticity may be degenerate. The study of the Kato conjecture for degenerate elliptic equations was begun by Cruz-Uribe and Rios (2008, 2012, 2015). In these papers the authors proved that given an operator , where is in the Muckenhoupt class and is a -degenerate elliptic measure (that is, with an bounded, complex-valued, uniformly elliptic matrix), then satisfies the weighted estimate . In the present paper we solve the -Kato problem for a family of degenerate elliptic operators. We prove that under some additional conditions on the weight , the following unweighted -Kato estimates hold:
This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator to have some degree of degeneracy in its ellipticity. For example, we consider the family of operators , where is any bounded, complex-valued, uniformly elliptic matrix. We prove that there exists , depending only on dimension and the ellipticity constants, such that
The case corresponds to the case of uniformly elliptic matrices. Hence, our result gives a range of ’s for which the classical Kato square root proved in Auscher et al. (2002) is an interior point.
Our main results are obtained as a consequence of a rich Calderón–Zygmund theory developed for certain operators naturally associated with . These results, which are of independent interest, establish estimates on , and also on with , for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and vertical square functions. As an application, we solve some unweighted -Dirichlet, regularity and Neumann boundary value problems for degenerate elliptic operators.
In the whole space , linear estimates for heat semigroup in Besov spaces are well established, which are estimates of type, with maximal regularity, etc. This paper is concerned with such estimates for the semigroup generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of .
Under certain hypotheses of smallness on the regular potential , we prove that the Dirac operator in , coupled with a suitable rescaling of , converges in the strong resolvent sense to the Hamiltonian coupled with a -shell potential supported on , a bounded surface. Nevertheless, the coupling constant depends nonlinearly on the potential ; Klein’s paradox comes into play.
An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multidimensional orthogonal expansions on product spaces. This is then applied to obtain boundedness, , of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the norms of these Riesz transforms are both dimension-free and linear in . The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.
We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimension with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree 2 in with coefficients which depend quasiperiodically on time.
We study the relationship between growth of eigenfunctions and their concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.
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